2nthrt (problem 3.4.6)

Average Error: 33.1 → 7.5

Time: 26.7s

Precision: binary64

Cost: 85644

Math

\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

↓

\[\begin{array}{l} \mathbf{if}\;x \leq 5.8 \cdot 10^{-171}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 8 \cdot 10^{-159}:\\ \;\;\;\;\frac{\frac{1 - {x}^{\left(\frac{4}{n}\right)}}{1 + {x}^{\left(\frac{2}{n}\right)}}}{1 + {x}^{\left(\frac{1}{n}\right)}}\\ \mathbf{elif}\;x \leq 3.1:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2}}{n \cdot n}, \frac{0.16666666666666666 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right)}{{n}^{3}}\right) + \left(\frac{{\log x}^{2}}{n \cdot n} \cdot -0.5 - \frac{\mathsf{fma}\left(-1, \mathsf{log1p}\left(x\right), \log x\right)}{n}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\frac{\log x}{n}}}{x \cdot n}\\ \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))

↓

(FPCore (x n)
 :precision binary64
 (if (<= x 5.8e-171)
   (/ (- (log x)) n)
   (if (<= x 8e-159)
     (/
      (/ (- 1.0 (pow x (/ 4.0 n))) (+ 1.0 (pow x (/ 2.0 n))))
      (+ 1.0 (pow x (/ 1.0 n))))
     (if (<= x 3.1)
       (+
        (fma
         0.5
         (/ (pow (log1p x) 2.0) (* n n))
         (/
          (* 0.16666666666666666 (- (pow (log1p x) 3.0) (pow (log x) 3.0)))
          (pow n 3.0)))
        (-
         (* (/ (pow (log x) 2.0) (* n n)) -0.5)
         (/ (fma -1.0 (log1p x) (log x)) n)))
       (/ (exp (/ (log x) n)) (* x n))))))

C

double code(double x, double n) {
	return pow((x + 1.0), (1.0 / n)) - pow(x, (1.0 / n));
}

↓

double code(double x, double n) {
	double tmp;
	if (x <= 5.8e-171) {
		tmp = -log(x) / n;
	} else if (x <= 8e-159) {
		tmp = ((1.0 - pow(x, (4.0 / n))) / (1.0 + pow(x, (2.0 / n)))) / (1.0 + pow(x, (1.0 / n)));
	} else if (x <= 3.1) {
		tmp = fma(0.5, (pow(log1p(x), 2.0) / (n * n)), ((0.16666666666666666 * (pow(log1p(x), 3.0) - pow(log(x), 3.0))) / pow(n, 3.0))) + (((pow(log(x), 2.0) / (n * n)) * -0.5) - (fma(-1.0, log1p(x), log(x)) / n));
	} else {
		tmp = exp((log(x) / n)) / (x * n);
	}
	return tmp;
}

Julia

function code(x, n)
	return Float64((Float64(x + 1.0) ^ Float64(1.0 / n)) - (x ^ Float64(1.0 / n)))
end

↓

function code(x, n)
	tmp = 0.0
	if (x <= 5.8e-171)
		tmp = Float64(Float64(-log(x)) / n);
	elseif (x <= 8e-159)
		tmp = Float64(Float64(Float64(1.0 - (x ^ Float64(4.0 / n))) / Float64(1.0 + (x ^ Float64(2.0 / n)))) / Float64(1.0 + (x ^ Float64(1.0 / n))));
	elseif (x <= 3.1)
		tmp = Float64(fma(0.5, Float64((log1p(x) ^ 2.0) / Float64(n * n)), Float64(Float64(0.16666666666666666 * Float64((log1p(x) ^ 3.0) - (log(x) ^ 3.0))) / (n ^ 3.0))) + Float64(Float64(Float64((log(x) ^ 2.0) / Float64(n * n)) * -0.5) - Float64(fma(-1.0, log1p(x), log(x)) / n)));
	else
		tmp = Float64(exp(Float64(log(x) / n)) / Float64(x * n));
	end
	return tmp
end

Wolfram

code[x_, n_] := N[(N[Power[N[(x + 1.0), $MachinePrecision], N[(1.0 / n), $MachinePrecision]], $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

↓

code[x_, n_] := If[LessEqual[x, 5.8e-171], N[((-N[Log[x], $MachinePrecision]) / n), $MachinePrecision], If[LessEqual[x, 8e-159], N[(N[(N[(1.0 - N[Power[x, N[(4.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[Power[x, N[(2.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 3.1], N[(N[(0.5 * N[(N[Power[N[Log[1 + x], $MachinePrecision], 2.0], $MachinePrecision] / N[(n * n), $MachinePrecision]), $MachinePrecision] + N[(N[(0.16666666666666666 * N[(N[Power[N[Log[1 + x], $MachinePrecision], 3.0], $MachinePrecision] - N[Power[N[Log[x], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[n, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[Power[N[Log[x], $MachinePrecision], 2.0], $MachinePrecision] / N[(n * n), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision] - N[(N[(-1.0 * N[Log[1 + x], $MachinePrecision] + N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Exp[N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] / N[(x * n), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < 5.7999999999999997e-171

   1. Initial program 42.6

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   2. Taylor expanded in n around inf 18.3

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]

   3. Simplified 18.3

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]
      Proof

      [Start] 18.3	\[ \frac{\log \left(1 + x\right) - \log x}{n} \]
      log1p-def [=>] 18.3	\[ \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

   4. Taylor expanded in x around 0 18.3

      \[\leadsto \frac{\color{blue}{-1 \cdot \log x}}{n} \]

   5. Simplified 18.3

      \[\leadsto \frac{\color{blue}{-\log x}}{n} \]
      Proof

      [Start] 18.3	\[ \frac{-1 \cdot \log x}{n} \]
      mul-1-neg [=>] 18.3	\[ \frac{\color{blue}{-\log x}}{n} \]

   if 5.7999999999999997e-171 < x < 7.99999999999999991e-159

   1. Initial program 47.6

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   2. Taylor expanded in x around 0 47.6

      \[\leadsto \color{blue}{1} - {x}^{\left(\frac{1}{n}\right)} \]

   3. Applied egg-rr 47.5

      \[\leadsto \color{blue}{\left(1 - {x}^{\left(2 \cdot {n}^{-1}\right)}\right) \cdot \frac{1}{1 + {x}^{\left({n}^{-1}\right)}}} \]

   4. Simplified 47.5

      \[\leadsto \color{blue}{\frac{1 - {x}^{\left(\frac{2}{n}\right)}}{1 + {x}^{\left(\frac{1}{n}\right)}}} \]
      Proof

      [Start] 47.5	\[ \left(1 - {x}^{\left(2 \cdot {n}^{-1}\right)}\right) \cdot \frac{1}{1 + {x}^{\left({n}^{-1}\right)}} \]
      associate-*r/ [=>] 47.5	\[ \color{blue}{\frac{\left(1 - {x}^{\left(2 \cdot {n}^{-1}\right)}\right) \cdot 1}{1 + {x}^{\left({n}^{-1}\right)}}} \]
      *-rgt-identity [=>] 47.5	\[ \frac{\color{blue}{1 - {x}^{\left(2 \cdot {n}^{-1}\right)}}}{1 + {x}^{\left({n}^{-1}\right)}} \]
      unpow-1 [=>] 47.5	\[ \frac{1 - {x}^{\left(2 \cdot \color{blue}{\frac{1}{n}}\right)}}{1 + {x}^{\left({n}^{-1}\right)}} \]
      associate-*r/ [=>] 47.5	\[ \frac{1 - {x}^{\color{blue}{\left(\frac{2 \cdot 1}{n}\right)}}}{1 + {x}^{\left({n}^{-1}\right)}} \]
      metadata-eval [=>] 47.5	\[ \frac{1 - {x}^{\left(\frac{\color{blue}{2}}{n}\right)}}{1 + {x}^{\left({n}^{-1}\right)}} \]
      unpow-1 [=>] 47.5	\[ \frac{1 - {x}^{\left(\frac{2}{n}\right)}}{1 + {x}^{\color{blue}{\left(\frac{1}{n}\right)}}} \]

   5. Applied egg-rr 47.5

      \[\leadsto \frac{\color{blue}{\left(1 - {x}^{\left(2 \cdot \frac{2}{n}\right)}\right) \cdot \frac{1}{1 + {x}^{\left(\frac{2}{n}\right)}}}}{1 + {x}^{\left(\frac{1}{n}\right)}} \]

   6. Simplified 47.5

      \[\leadsto \frac{\color{blue}{\frac{1 - {x}^{\left(\frac{4}{n}\right)}}{1 + {x}^{\left(\frac{2}{n}\right)}}}}{1 + {x}^{\left(\frac{1}{n}\right)}} \]
      Proof

      [Start] 47.5	\[ \frac{\left(1 - {x}^{\left(2 \cdot \frac{2}{n}\right)}\right) \cdot \frac{1}{1 + {x}^{\left(\frac{2}{n}\right)}}}{1 + {x}^{\left(\frac{1}{n}\right)}} \]
      /-rgt-identity [<=] 47.5	\[ \frac{\color{blue}{\frac{1 - {x}^{\left(2 \cdot \frac{2}{n}\right)}}{1}} \cdot \frac{1}{1 + {x}^{\left(\frac{2}{n}\right)}}}{1 + {x}^{\left(\frac{1}{n}\right)}} \]
      associate-/r/ [<=] 47.5	\[ \frac{\color{blue}{\frac{1 - {x}^{\left(2 \cdot \frac{2}{n}\right)}}{\frac{1}{\frac{1}{1 + {x}^{\left(\frac{2}{n}\right)}}}}}}{1 + {x}^{\left(\frac{1}{n}\right)}} \]
      remove-double-div [=>] 47.5	\[ \frac{\frac{1 - {x}^{\left(2 \cdot \frac{2}{n}\right)}}{\color{blue}{1 + {x}^{\left(\frac{2}{n}\right)}}}}{1 + {x}^{\left(\frac{1}{n}\right)}} \]
      div-sub [=>] 47.5	\[ \frac{\color{blue}{\frac{1}{1 + {x}^{\left(\frac{2}{n}\right)}} - \frac{{x}^{\left(2 \cdot \frac{2}{n}\right)}}{1 + {x}^{\left(\frac{2}{n}\right)}}}}{1 + {x}^{\left(\frac{1}{n}\right)}} \]
      pow-sqr [<=] 47.5	\[ \frac{\frac{1}{1 + {x}^{\left(\frac{2}{n}\right)}} - \frac{\color{blue}{{x}^{\left(\frac{2}{n}\right)} \cdot {x}^{\left(\frac{2}{n}\right)}}}{1 + {x}^{\left(\frac{2}{n}\right)}}}{1 + {x}^{\left(\frac{1}{n}\right)}} \]
      sqr-neg [<=] 47.5	\[ \frac{\frac{1}{1 + {x}^{\left(\frac{2}{n}\right)}} - \frac{\color{blue}{\left(-{x}^{\left(\frac{2}{n}\right)}\right) \cdot \left(-{x}^{\left(\frac{2}{n}\right)}\right)}}{1 + {x}^{\left(\frac{2}{n}\right)}}}{1 + {x}^{\left(\frac{1}{n}\right)}} \]
      div-sub [<=] 47.5	\[ \frac{\color{blue}{\frac{1 - \left(-{x}^{\left(\frac{2}{n}\right)}\right) \cdot \left(-{x}^{\left(\frac{2}{n}\right)}\right)}{1 + {x}^{\left(\frac{2}{n}\right)}}}}{1 + {x}^{\left(\frac{1}{n}\right)}} \]
      sqr-neg [=>] 47.5	\[ \frac{\frac{1 - \color{blue}{{x}^{\left(\frac{2}{n}\right)} \cdot {x}^{\left(\frac{2}{n}\right)}}}{1 + {x}^{\left(\frac{2}{n}\right)}}}{1 + {x}^{\left(\frac{1}{n}\right)}} \]
      pow-sqr [=>] 47.5	\[ \frac{\frac{1 - \color{blue}{{x}^{\left(2 \cdot \frac{2}{n}\right)}}}{1 + {x}^{\left(\frac{2}{n}\right)}}}{1 + {x}^{\left(\frac{1}{n}\right)}} \]
      associate-*r/ [=>] 47.5	\[ \frac{\frac{1 - {x}^{\color{blue}{\left(\frac{2 \cdot 2}{n}\right)}}}{1 + {x}^{\left(\frac{2}{n}\right)}}}{1 + {x}^{\left(\frac{1}{n}\right)}} \]
      metadata-eval [=>] 47.5	\[ \frac{\frac{1 - {x}^{\left(\frac{\color{blue}{4}}{n}\right)}}{1 + {x}^{\left(\frac{2}{n}\right)}}}{1 + {x}^{\left(\frac{1}{n}\right)}} \]

   if 7.99999999999999991e-159 < x < 3.10000000000000009

   1. Initial program 52.2

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   2. Taylor expanded in n around -inf 7.7

      \[\leadsto \color{blue}{\left(0.5 \cdot \frac{{\log \left(1 + x\right)}^{2}}{{n}^{2}} + \left(-1 \cdot \frac{-0.16666666666666666 \cdot {\log \left(1 + x\right)}^{3} - -0.16666666666666666 \cdot {\log x}^{3}}{{n}^{3}} + -1 \cdot \frac{-1 \cdot \log \left(1 + x\right) - -1 \cdot \log x}{n}\right)\right) - 0.5 \cdot \frac{{\log x}^{2}}{{n}^{2}}} \]

   3. Simplified 7.7

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2}}{n \cdot n}, \frac{0.16666666666666666 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right)}{{n}^{3}}\right) - \left(\frac{\mathsf{fma}\left(-1, \mathsf{log1p}\left(x\right), \log x\right)}{n} - \frac{{\log x}^{2}}{n \cdot n} \cdot -0.5\right)} \]
      Proof

      [Start] 7.7	\[ \left(0.5 \cdot \frac{{\log \left(1 + x\right)}^{2}}{{n}^{2}} + \left(-1 \cdot \frac{-0.16666666666666666 \cdot {\log \left(1 + x\right)}^{3} - -0.16666666666666666 \cdot {\log x}^{3}}{{n}^{3}} + -1 \cdot \frac{-1 \cdot \log \left(1 + x\right) - -1 \cdot \log x}{n}\right)\right) - 0.5 \cdot \frac{{\log x}^{2}}{{n}^{2}} \]

   if 3.10000000000000009 < x

   1. Initial program 21.4

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   2. Taylor expanded in x around inf 1.8

      \[\leadsto \color{blue}{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n \cdot x}} \]

   3. Simplified 1.8

      \[\leadsto \color{blue}{\frac{e^{\frac{\log x}{n}}}{x \cdot n}} \]
      Proof

      [Start] 1.8	\[ \frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n \cdot x} \]
      mul-1-neg [=>] 1.8	\[ \frac{e^{\color{blue}{-\frac{\log \left(\frac{1}{x}\right)}{n}}}}{n \cdot x} \]
      log-rec [=>] 1.8	\[ \frac{e^{-\frac{\color{blue}{-\log x}}{n}}}{n \cdot x} \]
      mul-1-neg [<=] 1.8	\[ \frac{e^{-\frac{\color{blue}{-1 \cdot \log x}}{n}}}{n \cdot x} \]
      distribute-neg-frac [=>] 1.8	\[ \frac{e^{\color{blue}{\frac{--1 \cdot \log x}{n}}}}{n \cdot x} \]
      mul-1-neg [=>] 1.8	\[ \frac{e^{\frac{-\color{blue}{\left(-\log x\right)}}{n}}}{n \cdot x} \]
      remove-double-neg [=>] 1.8	\[ \frac{e^{\frac{\color{blue}{\log x}}{n}}}{n \cdot x} \]
      *-commutative [=>] 1.8	\[ \frac{e^{\frac{\log x}{n}}}{\color{blue}{x \cdot n}} \]

3. Recombined 4 regimes into one program.

4. Final simplification 7.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5.8 \cdot 10^{-171}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 8 \cdot 10^{-159}:\\ \;\;\;\;\frac{\frac{1 - {x}^{\left(\frac{4}{n}\right)}}{1 + {x}^{\left(\frac{2}{n}\right)}}}{1 + {x}^{\left(\frac{1}{n}\right)}}\\ \mathbf{elif}\;x \leq 3.1:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2}}{n \cdot n}, \frac{0.16666666666666666 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right)}{{n}^{3}}\right) + \left(\frac{{\log x}^{2}}{n \cdot n} \cdot -0.5 - \frac{\mathsf{fma}\left(-1, \mathsf{log1p}\left(x\right), \log x\right)}{n}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\frac{\log x}{n}}}{x \cdot n}\\ \end{array} \]

Alternatives

Alternative 1
Error	7.7
Cost	20744

\[\begin{array}{l} \mathbf{if}\;x \leq 5.8 \cdot 10^{-171}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 8 \cdot 10^{-159}:\\ \;\;\;\;\frac{\frac{1 - {x}^{\left(\frac{4}{n}\right)}}{1 + {x}^{\left(\frac{2}{n}\right)}}}{1 + {x}^{\left(\frac{1}{n}\right)}}\\ \mathbf{elif}\;x \leq 3.1:\\ \;\;\;\;\frac{-1}{n} \cdot \log \left(\frac{x}{x + 1}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\frac{\log x}{n}}}{x \cdot n}\\ \end{array} \]

Alternative 2
Error	7.7
Cost	20616

\[\begin{array}{l} \mathbf{if}\;x \leq 5.8 \cdot 10^{-171}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 8 \cdot 10^{-159}:\\ \;\;\;\;\frac{1 - {x}^{\left(\frac{3}{n}\right)}}{{x}^{\left(\frac{2}{n}\right)} + \left(1 + {x}^{\left(\frac{1}{n}\right)}\right)}\\ \mathbf{elif}\;x \leq 3.1:\\ \;\;\;\;\frac{-1}{n} \cdot \log \left(\frac{x}{x + 1}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\frac{\log x}{n}}}{x \cdot n}\\ \end{array} \]

Alternative 3
Error	7.7
Cost	19912

\[\begin{array}{l} \mathbf{if}\;x \leq 5.8 \cdot 10^{-171}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 8 \cdot 10^{-159}:\\ \;\;\;\;{\left(\sqrt[3]{1 - {x}^{\left(\frac{1}{n}\right)}}\right)}^{3}\\ \mathbf{elif}\;x \leq 1.65:\\ \;\;\;\;\frac{-1}{n} \cdot \log \left(\frac{x}{x + 1}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\frac{\log x}{n}}}{x \cdot n}\\ \end{array} \]

Alternative 4
Error	7.7
Cost	13644

\[\begin{array}{l} \mathbf{if}\;x \leq 5.8 \cdot 10^{-171}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 8 \cdot 10^{-159}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 3.1:\\ \;\;\;\;\frac{-1}{n} \cdot \log \left(\frac{x}{x + 1}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\frac{\log x}{n}}}{x \cdot n}\\ \end{array} \]

Alternative 5
Error	23.7
Cost	8604

\[\begin{array}{l} t_0 := \frac{\frac{1}{n}}{x}\\ t_1 := \frac{-\log x}{n}\\ \mathbf{if}\;\frac{1}{n} \leq -2000000:\\ \;\;\;\;\frac{0}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq -1.7 \cdot 10^{-70}:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-270}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-237}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-184}:\\ \;\;\;\;\frac{\left(\frac{1}{x} + \frac{1}{x} \cdot \frac{0.3333333333333333}{x \cdot x}\right) + \frac{-0.5}{x \cdot x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-156}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-11}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]

Alternative 6
Error	14.8
Cost	7833

\[\begin{array}{l} t_0 := \frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{if}\;n \leq -5.3 \cdot 10^{+225}:\\ \;\;\;\;\frac{-1}{n} \cdot \log \left(\frac{x}{x + 1}\right)\\ \mathbf{elif}\;n \leq -1.3 \cdot 10^{+132}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{elif}\;n \leq -2.6 \cdot 10^{+110}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq -5.8 \cdot 10^{+69}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \mathbf{elif}\;n \leq -9.796324147911316 \cdot 10^{-308} \lor \neg \left(n \leq 17000000000\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]

Alternative 7
Error	14.9
Cost	7641

\[\begin{array}{l} t_0 := \frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{if}\;n \leq -1.85 \cdot 10^{+225}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq -4.2 \cdot 10^{+130}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{elif}\;n \leq -1.36 \cdot 10^{+110}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq -5.8 \cdot 10^{+69}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \mathbf{elif}\;n \leq 2.47957898479583 \cdot 10^{-301} \lor \neg \left(n \leq 2800000000\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]

Alternative 8
Error	14.9
Cost	7641

\[\begin{array}{l} t_0 := \frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{if}\;n \leq -7.4 \cdot 10^{+224}:\\ \;\;\;\;\frac{-1}{n} \cdot \log \left(\frac{x}{x + 1}\right)\\ \mathbf{elif}\;n \leq -2.4 \cdot 10^{+132}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{elif}\;n \leq -5.2 \cdot 10^{+107}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq -6 \cdot 10^{+69}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \mathbf{elif}\;n \leq -9.796324147911316 \cdot 10^{-308} \lor \neg \left(n \leq 4100000000\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]

Alternative 9
Error	16.3
Cost	6852

\[\begin{array}{l} t_0 := \frac{\frac{1}{n}}{x}\\ t_1 := \frac{0.3333333333333333}{x \cdot x}\\ \mathbf{if}\;x \leq 0.86:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 2.35 \cdot 10^{+89}:\\ \;\;\;\;\frac{\left(\frac{1}{x} + \frac{1}{x} \cdot t_1\right) + \frac{-0.5}{x \cdot x}}{n}\\ \mathbf{elif}\;x \leq 7.8 \cdot 10^{+119}:\\ \;\;\;\;\frac{0}{n}\\ \mathbf{elif}\;x \leq 7.8 \cdot 10^{+178}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot t_1\\ \end{array} \]

Alternative 10
Error	16.5
Cost	6788

\[\begin{array}{l} t_0 := \frac{\frac{1}{n}}{x}\\ t_1 := \frac{0.3333333333333333}{x \cdot x}\\ \mathbf{if}\;x \leq 0.6:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 2.26 \cdot 10^{+89}:\\ \;\;\;\;\frac{\left(\frac{1}{x} + \frac{1}{x} \cdot t_1\right) + \frac{-0.5}{x \cdot x}}{n}\\ \mathbf{elif}\;x \leq 1.16 \cdot 10^{+120}:\\ \;\;\;\;\frac{0}{n}\\ \mathbf{elif}\;x \leq 9 \cdot 10^{+178}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot t_1\\ \end{array} \]

Alternative 11
Error	34.3
Cost	964

\[\begin{array}{l} t_0 := \frac{\frac{1}{n}}{x}\\ \mathbf{if}\;\frac{1}{n} \leq -1:\\ \;\;\;\;t_0 \cdot \frac{0.3333333333333333}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 12
Error	32.9
Cost	964

\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1:\\ \;\;\;\;\frac{\frac{1}{x} \cdot \frac{0.3333333333333333}{x \cdot x}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \end{array} \]

Alternative 13
Error	35.8
Cost	841

\[\begin{array}{l} \mathbf{if}\;n \leq -0.9 \lor \neg \left(n \leq -1 \cdot 10^{-188}\right):\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(1 + \frac{1}{x \cdot n}\right)\\ \end{array} \]

Alternative 14
Error	28.7
Cost	585

\[\begin{array}{l} \mathbf{if}\;n \leq -0.9 \lor \neg \left(n \leq 5 \cdot 10^{-104}\right):\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{n}\\ \end{array} \]

Alternative 15
Error	40.7
Cost	320

\[\frac{1}{x \cdot n} \]

Alternative 16
Error	40.2
Cost	320

\[\frac{\frac{1}{n}}{x} \]

Reproduce

herbie shell --seed 2023060 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))